Q. When a string is divided into three segments of length $\mathcal{l}_{1},\mathcal{l}_{2}$ and $\mathcal{l}_{3}$ the fundamental frequencies of these three segments are $v_{1},v_{2}$ and $v_{3}$ respectively. The original fundamental frequency $(v)$ of the string is
A) $\sqrt{v} = \sqrt{v_{1}} + \sqrt{v_{2}} + \sqrt{v_{3}}$
B) $v = v_{1} + v_{2} + v_{3}$
C) $\frac{1}{v} = \frac{1}{v_{1}} + \frac{1}{v_{2}} + \frac{1}{v_{3}}$
D) $\frac{1}{\sqrt{v}} = \frac{1}{\sqrt{v_{1}}} + \frac{1}{\sqrt{v_{2}}} + \frac{1}{\sqrt{v_{3}}}$
Explanation
(c) Fundamental frequency is given by
$v = \frac{1}{2\mathcal{l}}\sqrt{\frac{T}{\mu}} \Rightarrow v \propto \frac{1}{\mathcal{l}}$
Here $\mathcal{l =}\mathcal{l}_{1} + \mathcal{l}_{2} + \mathcal{l}_{3}$
So $\frac{1}{v} = \frac{1}{v_{1}} + \frac{1}{v_{2}} + \frac{1}{v_{3}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.